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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 682436, 7 pages
A Regularity Criterion for the Navier-Stokes Equations in the Multiplier Spaces
1College of Physics and Electronic Information Engineering, Wenzhou University, Zhejiang, Wenzhou 325035, China
2The Key Laboratory of Low-voltage Apparatus Intellectual Technology of Zhejiang, Wenzhou 325035, China
Received 16 February 2012; Accepted 23 April 2012
Academic Editor: Benchawan Wiwatanapataphee
Copyright © 2012 Xiang'ou Zhu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
We exhibit a regularity condition concerning the pressure gradient for the Navier-Stokes equations in a special class. It is shown that if the pressure gradient belongs to , where is the multipliers between Sobolev spaces whose definition is given later for , then the Leray-Hopf weak solution to the Navier-Stokes equations is actually regular.
Consider the Navier-Stokes equations in : where is the velocity field, is the scalar pressure, and with div in the sense of distribution is the initial velocity field. For simplicity, we assume that the external force has a scalar potential and is included into the pressure gradient.
In the famous paper, Leray  and Hopf  constructed a weak solution of (1.1) for arbitrary with . The solution is called the Leray-Hopf weak solution. Regularity of such Leray-Hopf weak solutions is one of the most significant open problems in mathematical fluid mechanics. We note here that there are partial regularity results from Scheffer and from Caffarelli et al., see [3, 4] and references therein. Besides, more work was pioneered by Serrin  and extended and improved by Giga , Struwe [7, 8], and Zhou . Further results can be found in [10–16] and references therein.
Introducing the class , Serrin  showed that if we have a Leray-Hopf weak solution belonging to with the exponents and satisfying , , , then the solution , while the limit case was shown much later by Sohr  (see also ).
Regularity results including assumptions on the pressure gradient have been given by Zhou , and it was extended later by Struwe  to any dimension . It is shown that if the gradient of pressure with , then the corresponding weak solution is actually strong. For the recent work on the regularity problem containing the pressure, velocity field, and the quotient of pressure-velocity, we refer to [19–21] for details.
The purpose of this short paper is to establish a regularity criterion in terms of the pressure gradient for weak solutions to the Navier-Stokes equations in the class . This work is motivated by the recent results [22, 23] on the Navier-Stokes equations. It is an unusual, larger space considered in the current paper than (the following Lemma 2.3) and possesses more information. Obviously, the present result extends some previous ones. For more facts concerning regularity of weak solutions, we refer the readers to the celebrated papers [24–30].
Definition 2.1. For , is a Banach space of all distributions on such that there exists a constant such that for all we have and where we denote by the completion of the space with respect to the norm and denote by the Schwarz class.
The norm of is given by the operator norm of pointwise multiplication
Remark 2.2. Equivalently, we will say that if and only if the inequality holds for all .
Lemma 2.3. Let . Then the following embedding: holds.
Proof. Indeed, let . By using the following well-known Sobolev embedding: with , we have by Hölder’s inequality where . Then, it follows that This completes the proof.
Example 2.4. Due to the well-known inequality we see that .
Indeed, since the functions of class are dense in in the norm , suppose . Then by virtue of the Cauchy-Schwarz inequality we obtain and thus for , and
3. Regularity Theorem
Now we state our result as following.
Theorem 3.1. Let for some and in the sense of distributions. Suppose that is a Leray-Hopf solution of (1.1) in . If the pressure gradient satisfies then is a regular solution in the sense that
Proof. In order to prove this result, we have to do a priori estimates for the Navier-Stokes equations and then show that the solution satisfies the well-known Serrin regularity condition. Multiply both sides of the first equation of (1.1) by and integrate by parts to obtain (see, e.g., )
for . Then we have
where we have used
Let us estimate the integral on the right-hand side of (3.4). By the Hölder inequality and the Young inequality, we have where ; we have used the inequality and the Young inequality with : for . Hence by (3.4) and the above inequality, we derive Now by Gronwall’s lemma (see for instance in [28, Lemma 2]), we have Due to the integrability of the pressure gradient, it follows that Consequently falls into the well-known Serrin’s regularity framework. Therefore, the smoothness of follows immediately. This completes the proof of Theorem 3.1.
Remark 3.2. By a strong solution we mean a weak solution of the Navier-Stokes equation such that It is wellknown that strong solutions are regular (we say classical) and unique in the class of weak solutions.
The author thanks the anonymous referee for his/her comments on this paper.
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